Pure dephasing in flux qubits due to flux noise with spectral density scaling as $1/ f^\alpha$

TL;DR

This paper investigates how flux noise with a $1/f^eta$ spectral density affects the pure dephasing times of superconducting flux qubits, revealing that the spectral exponent significantly influences qubit coherence.

Contribution

It provides a detailed analysis of the impact of flux noise spectral scaling on qubit dephasing, including the dependence on the spectral exponent and the cutoff frequencies, which was not previously characterized.

Findings

Dephasing time decreases rapidly as the spectral exponent $eta$ decreases.

Dephasing times are relatively insensitive to the noise bandwidth if the ultraviolet cutoff exceeds $1/\tau_\phi$.

The choice of pivot frequency $f_0$ significantly affects the sensitivity of dephasing times to the spectral exponent.

Abstract

For many types of superconducting qubits, magnetic flux noise is a source of pure dephasing. Measurements on a representative dc superconducting quantum interference device (SQUID) over a range of temperatures show that $S_\Phi(f) = A^2/(f/1 \hbox{Hz})^\alpha$, where $S_\Phi$ is the flux noise spectral density, $A$ is of the order of 1 $\mu\Phi_0 \, \hbox{Hz}^{-1/2}$ and $0.61 \leq \alpha \leq 0.95$; $\Phi_{0}$ is the flux quantum. For a qubit with an energy level splitting linearly coupled to the applied flux, calculations of the dependence of the pure dephasing time $\tau_\phi$ of Ramsey and echo pulse sequences on $\alpha$ for fixed $A$ show that $\tau_\phi$ decreases rapidly as $\alpha$ is reduced. We find that $\tau_\phi$ is relatively insensitive to the noise bandwidth, $f_1 \leq f \leq f_2$, for all $\alpha$ provided the ultraviolet cutoff frequency $f_2 > 1/\tau_\phi$. We calculate the ratio $\tau_{\phi,E} / \tau_{\phi,R}$ of the echo ($E$) and Ramsey ($R$) sequences, and the dependence of the decay function on $\alpha$ and $f_2$. We investigate the case in which $S_\Phi(f_0)$ is fixed at the "pivot frequency" $f_0 \neq 1$ Hz while $\alpha$ is varied, and find that the choice of $f_0$ can greatly influence the sensitivity of $\tau_{\phi,E}$ and $\tau_{\phi,R}$ to the value of $\alpha$. Finally, we present calculated values of $\tau_\phi$ in a qubit corresponding to the values of $A$ and $\alpha$ measured in our SQUID.